Predictive Dynamics: Lifting - Human Motion Simulation: Predictive Dynamics - page 188

The derivatives @τ i

@q k , @τ i @ q k , @τ i

( i 5 1to n ; k 5 1to n ) are evaluated for the artic-

@ q k

ulated spatial human mechanical system in a recursive manner using the forego-
ing recursive Lagrangian dynamics formulation as follows:

<

tr @

2 A i 2 A i 2 A i

q k D i 1 @ A i q i @ D i

@ @

G i @ A i 2 1

g T f T

q k E i 2 q k F i 2

z 0 ðk iÞ

2 #

@ q i @ @ @ q k @ q i @ @ q i @ @ q k

@τ i
@

tr @

2 g T @

q k 5

:

@q i @

A i D i

@q k @q i @

A i

@q k 2 f T @

E i

@q i @

A i F i

@q k

ðk . iÞ

(8.10)

@τ i
@ _

tr @ A i
@ q i @ D i

(8.11)

q k 5

@ _ q k

@τ i
@ €

tr @

A i
@ q i @

D i
@ €

(8.12)

q k 5 q k

Note that the computational cost of the recursive formulation is of the order
OðnÞ , where n is the number of DOFs. Forward kinematics transfers the motion
from the origin toward the end-effector along the branch. In contrast, the backward
dynamics propagates forces from the end-effector to the origin. More details about
the derivation of sensitivity equations are provided by Xiang et al. (2009a,b) .

8.3 Dynamic stability and ground reaction forces (GRF)
We shall use the ZMP method to address the dynamic stability condition for the
lifting motion. We shall also include the GRF to be calculated. This can be
accomplished by forcing the feet to stay in the form of a polygon as shown in
Figure 8.2 . The two feet are fixed on the ground with the distance d and orienta-
tion angle

during the lifting motion. The concept of ZMP has been extensively
used as a bipedal dynamic stability criterion. It is defined as the point on the
ground at which the resultant tangential moments are zero.
We shall also use an active-passive algorithm to calculate ZMP and GRF to obtain
the real joint torques for the multi-body human system. Details of the algorithm are
presented by Xiang et al. (2009a,b) and Hariri (2012) and outlined as follows:
1. Use inverse dynamics to calculate the global resultant active forces, M o , F o ,
at the origin in the inertial reference frame (o-xyz in Figure 8.2 ). Note that the
state variables q ,

θ

q (design variables) are specified for each DOF.
2. Calculate the ZMP position from its definition using the global resultant
active forces as follows:

q ,

M z
F y ;

M x
F y

z zmp 5 2

y zmp 5 0 ; x zmp 5 (8.13)